May 01, 2017
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
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- Supplements
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09:30 AM - 10:30 AM
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Better than squareroot cancellation for multiplicative functions
Adam Harper (University of Warwick)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
It is a standard heuristic that sums of oscillating number theoretic functions, like the M\"obius function or Dirichlet characters, should exhibit squareroot cancellation. It is often very difficult to prove anything as strong as that, and we generally expect that if we could prove squareroot cancellation it would be the best possible bound. I will discuss recent results showing that, in fact, certain averages of multiplicative functions exhibit a bit more than squareroot cancellation
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Approximate cohomology
Tamar Ziegler (The Hebrew University of Jerusalem)
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- Location
- SLMath: Atrium
- Video
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- Abstract
Let V be an infinite vector space over a finite field k and let be an approximate homomorphism from V to End(V), i.e. the rank of f(x+y)-f(x)-f(y) is uniformly bounded by some constant r. We show that there is a homomorphism g such that f-g is of rank uniformly bounded by R(r).
We introduce a notion of a approximate cohomology groups and interpret this result as a computation of H^1 in a special case. Our proof uses the inverse theorem for the Gowers norms over finite fields; and an independent proof of this result (and some natural generalizations of it) would lead to a new proof of the inverse theorem. Joint work with D. Kazhdan
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Subconvex equidistribution of cusp forms
Paul Nelson (ETH Zürich)
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- Location
- --
- Video
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- Abstract
Arithmetic quantum chaos concerns the limiting behavior of a sequence of automorphic forms with parameters tending off to infinity. It is now known in many cases that the mass distributions of such forms equidistribute. Unfortunately, the known rates of equidistribution are typically weak (ineffective or logarithmic). I will discuss the problem of obtaining strong rates (power savings) and the related subconvexity problem, emphasizing recent progress concerning the level aspect on hyperbolic surfaces.
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Averages of p-torsion in class groups over function fields---good and bad primes
Melanie Wood (Harvard University; University of California, Berkeley)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
When we consider the average size of the p-torsion in class groups of quadratic fields, the behavior for p=2 is controlled by genus theory and is different from the conjectured behavior for odd primes, which is uniform in a certain sense over all odd primes. In this talk, we will consider the question when quadratic fields are replaced by fields of higher degree--for which primes p are the class group averages "bad" versus "good"? We will explain some theorems on class group averages for extensions of function fields that show some subtleties in the classification of good and bad primes
- Supplements
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May 02, 2017
Tuesday
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09:30 AM - 10:30 AM
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A Chebotarev density theorem for families of fields, with applications to class groups
Lillian Pierce (Duke University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
This talk will present a new effective Chebotarev theorem that holds for all but a possible zero-density subfamily of certain families of number fields of fixed degree. For certain families, this work is unconditional, and in other cases it is conditional on the strong Artin conjecture and certain conjectures on counting number fields. As a result, we obtain nontrivial average upper bounds on p-torsion in the class groups of the families of fields.
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Applications of Exponential Sums
Will Sawin (ETH Zürich)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We discuss recent work, including joint with with Emmanuel Kowalski and Philippe Michel, and possibly others, where new bounds on complete exponential sums were proved, which were then applied to analytic number theory problems. We will also discuss some general heuristics for how useful complete exponential sum estimates will be for a particular problem
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Poster Session
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Squarefree values of polynomial discriminants
Manjul Bhargava (Princeton University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
The question as to whether there exist a positive proportion of monic irreducible integral polynomials of degree n having squarefree discriminant is an old one; an exact formula for the density was conjectured by Lenstra. (The interest in monic irreducible polynomials f with squarefree discriminant comes from the fact that in such cases
Z[x]/(f(x)) gives the ring of integers in the number field Q[x]/(f(x)).)
In this talk, we describe recent work with Arul Shankar and Xiaoheng Wang that allows us to determine the probability that a random monic integer polynomial has squarefree discriminant - thus proving the conjecture of Lenstra.
- Supplements
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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May 03, 2017
Wednesday
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09:30 AM - 10:30 AM
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Correlations of von Mangoldt and higher order divisor functions
Kaisa Matomäki (University of Turku)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will discuss joint work with M. Radziwill and T. Tao on asymptotics for the sums $\sum_{n \leq x} \Lambda(n) \Lambda(n+h)$ and $\sum_{n \leq x} d_k(n) d_l(n+h)$ where $\Lambda$ is the von Mangoldt function and $d_k$ is the kth divisor function. For the first sum we show that the expected asymptotics hold for almost all $|h| \leq X^{8/33}$ and for the second sum we show that the expected asymptotics hold for almost all $|h| \leq (\log X)^{O_{k, l} ( 1) }$.
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Beyond Expansion and Arithmetic Chaos
Alex Kontorovich (Rutgers University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We will describe recent progress in our ongoing program with Jean Bourgain to understand a number of different problems through the lens of thin orbits. An important role will be played by the production of levels of distribution (in certain Affine Sieves) which go "beyond expansion." No prior knowledge of these topics is assumed
- Supplements
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May 04, 2017
Thursday
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09:30 AM - 10:30 AM
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On integral points on degree four del Pezzo surfaces
Damaris Schindler (Universiteit Utrecht)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We report on our investigations concerning algebraic and transcendental Brauer-Manin obstructions to integral points on complements of a hyperplane section in degree four del Pezzo surfaces. This is joint work with Joerg Jahnel
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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The fifth moment of modular L-functions
Matthew Young (Texas A & M University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will discuss recent joint work with E. Mehmet Kiral which bounds the fifth moment of L-functions of fixed small weight and large prime level
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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The orbit method and analysis of automorphic forms
Akshay Venkatesh (Institute for Advanced Study)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
In the analytic theory of automorphic forms, especially in higher rank, one encounters complicated integrals over Lie groups which must be either evaluated or estimated. I will discuss how Kirillov's orbit method allows one to do this, at least heuristically. These ideas can often be made rigorous; I will apply it to evaluate the average value of L-functions over certain (Gross-Prasad) families, in any rank. This evaluation also uses Ratner's theorem on measure rigidity. Joint work with Paul Nelson.
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Large gaps between primes in subsets
James Maynard (University of Oxford)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long strings of consecutive composite values of a polynomial. This is joint work with Ford, Konyagin, Pomerance and Tao
- Supplements
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May 05, 2017
Friday
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09:30 AM - 10:30 AM
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Arithmetic functions: something old, something new
Paul Pollack (University of Georgia)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will discuss recent joint work on the value distribution of arithmetic functions. The problems discussed have in common that they owe their origin --- in one way or another --- to the fascination of the ancients with sums of divisors
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Local to global principles in integral circle packings
Elena Fuchs (University of California, Davis)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
One of the most spectacular results on arithmetic of Apollonian circle packings is the "almost" local to global principle for curvatures in any given integral Apollonian packing as described by Bourgain-Kontorovich in 2014. The methods in their work, inspired originally by an observation of Sarnak’s in his letter to Lagarias on Apollonian circle packings, apply to a much larger class of circle packings. In this talk, we clarify what "almost" local to global means, and describe what the larger class is, as well as what aspects of the packings in this class seem necessary in order to conclude an "almost" local to global result and how they enter the proof. This is joint work with Stange and Zhang.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Bombieri-Vinogradov for general multiplicative functions
Fernando Shao (University of Oxford)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. In this talk I will discuss a Bombieri-Vinogradov type theorem for multiplicative functions and develop some limitations. This is joint work with Andrew Granville.
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Nested Efficient Congruencing and (non) translation-dilation invariance.
Trevor Wooley (University of Bristol)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We discuss the nested refinement of the efficient congruencing method, which incorporates an idea from the l^2-decoupling work of Bourgain, Demeter and Guth. There are consequences for (non) translation-dilation invariant systems
- Supplements
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